In this paper we extend an earlier worst case bound reliability theory to derive a worst case reliability function R(t), which gives the worst case probability of surviving a further time t given an estimate of residual defects in the software N and a prior test time T. The earlier theory and its extension are presented and the paper also considers the case where there is a low probability of any defect existing in the program. For the "fractional defect" case, there can be a high probability of surviving any subsequent time t. The implications of the theory are discussed and compared with alternative reliability models. Keywords: reliability prediction, reliability testing, worst case reliability bound, residual fault prediction.